Consider a continuous, positive function f: R --> R. The graph of f(x) is the set of points in the (x,y) Cartesian plane such that y = f(x).
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Please see the attachment.
We kow that the area can be represented as a definite integral as follows.
(a) Since , , then for any , we have . Then we have
(b) Set . From (a), we get . Since is a continuous function from the graph, then can reach maximum and minimum in the closed interval . Then we can find some , such that and . By the Intermediate Value Theorem, we can find some between and , such that . Then we have , where is between and , hence is in .
(c) We note that represent the area between and , according to (b), we can find some , such that . Therefore, we get .
(d) From part (c), we have , then we get
Let , then and the right side is the definition of . Then we have© BrainMass Inc. brainmass.com October 7, 2022, 5:06 pm ad1c9bdddf>